tight and relatively compact measures

Tight and relatively compact measuresFernando Sanz

Definition 1.

Let $ℳ=\{{\mu }_i,i\in I\}$ be a family of finite measures on theBorel subsets of a metric space $\mathrm{\Omega }$. We say that $ℳ$ is tightiff for each $ϵ>0$ there is a compact set $K$ such that for all $i$. We say that $ℳ$ isrelatively compact iff each sequence in $ℳ$ has a subsequenceconverging weakly to a finite measure on $ℬ(\mathrm{\Omega })$.

If $\{F_i,i\in I\}$ is a family of distribution functions,relative compactness or tightness of $\{F_i\}$ refers to relativecompactness or tightness of the corresponding measures.

Theorem.

Let $\mathrm{\{}{F}_i\mathrm{,}i\mathrm{\in }I\mathrm{\}}$ be a family of distribution functions with for all $i$. The family istight iff it is relatively compact.

Proof.

Coming soon…(needs other theorems before)∎